- Research Article
- Open Access
Sharpening the Becker-Stark Inequalities
© L. Zhu and J. Hua 2010
- Received: 3 April 2009
- Accepted: 14 January 2010
- Published: 20 January 2010
In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.
- Power Series
- Natural Number
- Series Expansion
- Zeta Function
- Power Series Expansion
Theorem 1.1 (see [1, Lemma ]).
Furthermore, and are the best constants in (1.2).
In fact, we can obtain the following further results.
Furthermore, and are the best constants in (1.3).
In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.
where are the even-indexed Bernoulli numbers.
Furthermore, and are the best constants in (1.4).
The function is decreasing, where is Riemann's zeta function.
Lemma 2.2 (see [5, Theorem ]).
Lemma 2.3 (see [6, 18.104.22.168 (1.3)]).
which implies that for .
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